On the other side, the equation \(Y^3h_{n}(X/Y)=\pm 1\)  associated with the simplest cubic fields of Shanks, defines a (parametric) Thue type equation and it is natural to ask about characterization of its integral solutions. In this direction one may consult the results [1]}, [2]}, [3]}, [4]}. For any given \(n\) , one can also consider the equation \(h_{n}(X)=Y^2\)  as the equation defining an elliptic curve and mention [5]}, where questions concerning the existence of integer/rational points, characterization of torsion groups and ranks are investigated. It is clear that analogous question can be asked to the more general family \((P_{f(n),g(n)})_{n\in \mathbb {N}}\) .
